In January, Melrose revealed it had made a £7.4bn takeover bid for GKN, an offer which was rejected by the GKN board.
According to the FTSE 100 firm, the new bid does not reflect the value of GKN's "world class aerospace business" or the benefits of combining GKN Driveline and Dana Incorporated - GKN last week agreed to merge its auto parts business with the USA firm in a deal worth $6.1bn (£4.4bn). "GKN Directors will do the same in respect of their own beneficial shareholdings". It said it study the new offer.
Melrose shares lost 3.4 percent to 217 pence, dragging down the value of the bid to 447 pence.
In an open letter to GKN shareholders, Melrose said the Dana deal is floored as many investors won't be able to hold the shares being offered by Dana as part of the deal.
Melrose said on Monday that the deal with Dana was "ill-thought-through" and would face regulatory hurdles.
Melrose also warned that private United Kingdom shareholders should also be aware that they would be liable to income tax on the value of any new Dana shares they receive and the transaction would "involve a lengthy and uncertain completion process, including anti-trust clearances in the EU, US and China, as well as Dana's shareholder approval, which is not expected until the last quarter of 2018". Its aerospace division has also attracted a number of suitors since January, but the GKN board decided their proposals undervalued the business and did not pursue them, the company added. It has offered to give back £2.5bn to shareholders and agreed to merge its auto unit with U.S. company Dana.
Melrose's potential takeover of GKN has prompted worries from British politicians about United Kingdom jobs being lost and the country's engineering know-how ending up in foreign hands.
Rebecca O'Keeffe, head of investment at Interactive Investor, said: "The muted market reaction of the GKN share price to the increased offer of 467p by Melrose is the strongest indication yet that Melrose might not get its way and that GKN's management and their Project Boost strategy is winning".
Lawmakers were concerned that Melrose could break up the engineer and sell parts to foreign buyers, potentially compromising British and US national security because of GKN's work on defence programmes - the company makes components for the Eurofighter Typhoon combat jet.
Shareholders have until March 29 to accept the offer, and thwart what it called a "hasty fire-sale" of GKN's businesses that would be a "bad deal".
"We are more convinced than ever that the Melrose team, who have decades of experience in successfully transforming businesses, are the only real choice of team to re-energise and re-focus GKN to unlock its full potential", Melrose said in its statement.